Let \({C}\)  be a small category, and let \({{K}}_\infty \)  denote the category of simplicial presheaves on \({C}\) .
We denote by \({{K}}_\infty ^{p/i}\)  the category \({{K}}_\infty \)  endowed with the projective or the injective model structure, respectively.
Let \({I}\)  be a small category, and let \(E \colon {I}\rightarrow {{K}}_\infty \)  be a diagram.
We can equivalently view \(E\)  as a functor \(E \colon {C}^\mathrm {op}\times {I}\rightarrow {{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\) .
Further, let \(F \colon {I}^\mathrm {op}\rightarrow {{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  be a functor.
Given these data, there is an associated two-sided bar construction [1]}
\(B^{{{K}}_\infty }(F, {I}, E)= \int ^{n \in {\mathbbe {\Delta }}} \Delta ^n \otimes B^{{{K}}_\infty }_n(F, {I}, E)\,,\) 
